Edge homogenization of Dyson Brownian motion and applications (2509.14192v1)

Abstract: We prove a homogenization result for the difference of two coupled Dyson Brownian motions started from generalized Wigner matrix initial data. We prove an optimal order, high probability estimate that is valid throughout the spectrum, including up to the spectral edges. Prior homogenization results concerned only the bulk of the spectrum. We apply our estimate to address the question of quantifying edge universality. Here, we have two results. We show that the Kolmogorov-Smirnov distance of the distribution of the gap between the largest two eigenvalues of a generalized Wigner matrix (with smooth entry distribution) and its GOE/GUE counterpart is $\mathcal{O}(N^{-1+\varepsilon})$. On the other hand, we show that, for the distribution of the largest eigenvalue, there are Wigner matrices so that the analogous Kolmogorov-Smirnov distance is bounded below by $N^{-1/3-\varepsilon}$.

• The paper establishes an optimal homogenization theory for Dyson Brownian motion at the spectral edge, providing sharp estimates that extend universality results beyond the bulk.
• The paper employs a refined iterative scheme and parabolic maximum principle to address divergent scales and accurately capture eigenvalue statistics.
• The paper demonstrates gap universality and the optimal convergence rate for the largest eigenvalue, affirming the sensitivity of edge behavior to non-Gaussian cumulants.

Edge Homogenization of Dyson Brownian Motion and Applications

Introduction and Motivation

This paper develops an optimal homogenization theory for Dyson Brownian motion (DBM) at the spectral edge, extending previous results that were limited to the bulk of the spectrum. Homogenization in DBM is a central technique for proving universality results in random matrix theory, particularly for Wigner matrices. The main technical achievement is the derivation of sharp estimates for the difference between two coupled DBMs, initialized from generalized Wigner matrices, valid throughout the entire spectrum—including the edge and intermediate regimes.

The motivation is twofold: (1) to provide a rigorous framework for quantitative edge universality, and (2) to address open questions regarding the rate of convergence of eigenvalue statistics (e.g., the largest eigenvalue and the gap between the top two eigenvalues) to their universal limits, such as the Tracy-Widom distribution.

Homogenization Theory for DBM at the Edge

The DBM describes the evolution of eigenvalues $\{x_i(t)\}_{i=1}^N$ of random matrices under a system of SDEs with repulsive interactions and a confining potential. Previous homogenization results, notably Bourgade [bourgade2018extreme], provided optimal estimates in the bulk but failed near the spectral edge due to diverging spatial and temporal scales.

The main result is a homogenization estimate for the difference $u_k(t) = e^{t/2}(\lambda_k(t) - \mu_k(t))$ between two coupled DBMs (with identical Brownian motions), initialized from two independent generalized Wigner matrices. The difference satisfies a long-range parabolic equation, and the leading order behavior is captured by a deterministic function $_k(t)$ involving the fundamental solution of the limiting PDE.

Theorem (Main Homogenization Estimate): For any $D>0$ and sufficiently large $N$,

$$P\left[ \left| \lambda_k(t) - \mu_k(t) - e^{-t/2} _k(t) \right| \leq \frac{N^\varepsilon}{N^2 t \rho_k (t+\rho_k)^2} \ \forall\ 1 \leq k \leq N,\ t \in (0,1) \right] \geq 1 - N^{-D}$$

where $_k(t)$ is explicitly constructed from the initial data and the semicircle quantiles.

The proof overcomes two main obstacles:

• Divergent scales at the edge: The natural spatial and temporal scales are $N^{-2/3}$ and $N^{-1/3}$, respectively, requiring a refined iterative scheme and careful cancellation of long-range and time-derivative error terms.
• Inhomogeneity due to semicircle curvature: The localization procedure is adapted to handle the non-uniform density near the edge, ensuring the iterative argument remains effective.

Quantitative Edge Universality

The homogenization theory enables new quantitative universality results for eigenvalue statistics at the edge. The paper proves two main results:

1. Gap Universality (Berry-Esseen Analog): For smooth generalized Wigner matrices,

$$\sup_{s \in \mathbb{R}} \left| P\left[ N^{2/3}(\lambda_1 - \lambda_2) \geq s \right] - P\left[ N^{2/3}(\mu_1 - \mu_2) \geq s \right] \right| \leq C_\varepsilon N^{-1+\varepsilon}$$

for all $\varepsilon > 0$, where $\mu_i$ are GOE eigenvalues. This matches the classical Berry-Esseen rate for sums of i.i.d. variables, but in the highly correlated setting of random matrices.

1. Optimality and Lower Bound for Largest Eigenvalue: For Wigner matrices with nonzero fourth cumulant,

$$\sup_{r \in \mathbb{R}} \left| P\left[ N^{2/3}(\lambda_1 - 2) \geq r \right] - P\left[ N^{2/3}(\mu_1 - 2) \geq r \right] \right| \geq c N^{-1/3+\varepsilon}$$

for any $\varepsilon > 0$. This demonstrates that the $N^{-1/3}$ rate obtained by Schnelli and Xu [schnelli2022convergence] is sharp in general, and that further improvement requires model-dependent shifts and scalings.

The gap statistic is shown to be more universal than the position of the largest eigenvalue, as the latter is sensitive to non-Gaussian cumulants.

Methodological Innovations

The technical core is a sophisticated iterative scheme based on the maximum principle for parabolic equations, spatial localization, and careful control of error terms. The analysis leverages:

• Rigidity and local laws for eigenvalue locations,
• Explicit formulas for the fundamental solution of the limiting PDE,
• Finite speed of propagation for short-range DBM approximations,
• Martingale concentration inequalities for stochastic terms.

The approach is robust and generalizes to other matrix ensembles, with potential for further extension to sample covariance matrices and sparse random matrices.

Implications and Future Directions

The results have several important implications:

• Universality at the edge is fundamentally different from the bulk: The optimal rate for the largest eigenvalue is $N^{-1/3}$, not $N^{-1}$, unless non-universal shifts are introduced.
• Eigenvalue gaps are more robustly universal: The gap between the top two eigenvalues converges at the Berry-Esseen rate, even in the presence of non-Gaussian cumulants.
• Homogenization throughout the spectrum: The new theory enables analysis of eigenvalue statistics at all scales, not just the bulk, opening the door to optimal results for linear spectral statistics, eigenvector statistics, and extreme value problems.

Potential future developments include:

• Extension of homogenization to other ensembles (e.g., sample covariance, sparse matrices),
• Improved resolvent methods for sharper rates below the $N^{-1}$ scale,
• Applications to spectral form factors, characteristic polynomial maxima, and CLTs for linear statistics at the edge.

Conclusion

This work establishes a comprehensive homogenization theory for Dyson Brownian motion at the spectral edge, yielding optimal quantitative universality results for eigenvalue statistics of Wigner matrices. The technical advances resolve longstanding questions about convergence rates and provide a foundation for further progress in random matrix theory and its applications to high-dimensional statistics and mathematical physics.